Properties

Label 1332.2.l.b
Level $1332$
Weight $2$
Character orbit 1332.l
Analytic conductor $10.636$
Analytic rank $0$
Dimension $74$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1332,2,Mod(121,1332)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1332.121"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1332, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1332 = 2^{2} \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1332.l (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [74] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.6360735492\)
Analytic rank: \(0\)
Dimension: \(74\)
Relative dimension: \(37\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 74 q + 5 q^{3} + 6 q^{5} - q^{7} - q^{9} + q^{11} - 4 q^{13} - 2 q^{15} + 3 q^{17} + q^{19} - 3 q^{21} - 17 q^{23} + 96 q^{25} + 17 q^{27} + 6 q^{29} + 3 q^{31} - 16 q^{33} - 6 q^{35} + 5 q^{37} + 35 q^{39}+ \cdots - 43 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1 0 −1.72977 − 0.0888169i 0 −2.43237 0 −0.249367 + 0.431916i 0 2.98422 + 0.307266i 0
121.2 0 −1.70009 + 0.331210i 0 2.31795 0 0.974750 − 1.68832i 0 2.78060 − 1.12617i 0
121.3 0 −1.65481 − 0.511467i 0 −3.76461 0 1.95890 − 3.39292i 0 2.47680 + 1.69276i 0
121.4 0 −1.64499 + 0.542232i 0 −2.30721 0 −1.48001 + 2.56345i 0 2.41197 − 1.78393i 0
121.5 0 −1.63291 + 0.577580i 0 3.36591 0 −2.09846 + 3.63465i 0 2.33280 − 1.88628i 0
121.6 0 −1.54587 − 0.781213i 0 2.77669 0 0.0407087 − 0.0705095i 0 1.77941 + 2.41530i 0
121.7 0 −1.46851 + 0.918407i 0 0.575944 0 1.60251 − 2.77564i 0 1.31306 − 2.69738i 0
121.8 0 −1.36126 − 1.07096i 0 2.16197 0 0.198964 − 0.344617i 0 0.706072 + 2.91573i 0
121.9 0 −1.11598 + 1.32461i 0 −1.89974 0 −0.786628 + 1.36248i 0 −0.509195 − 2.95647i 0
121.10 0 −1.05490 − 1.37375i 0 −1.33202 0 −2.25492 + 3.90564i 0 −0.774360 + 2.89834i 0
121.11 0 −1.02986 + 1.39262i 0 1.17070 0 0.566497 − 0.981202i 0 −0.878772 − 2.86841i 0
121.12 0 −0.647484 − 1.60648i 0 −1.93660 0 0.807677 − 1.39894i 0 −2.16153 + 2.08033i 0
121.13 0 −0.507120 − 1.65615i 0 3.15951 0 −2.24858 + 3.89466i 0 −2.48566 + 1.67973i 0
121.14 0 −0.400587 − 1.68509i 0 1.10274 0 2.25547 − 3.90659i 0 −2.67906 + 1.35005i 0
121.15 0 −0.340226 + 1.69831i 0 −2.74033 0 1.38459 − 2.39818i 0 −2.76849 − 1.15562i 0
121.16 0 −0.337501 − 1.69885i 0 −3.96562 0 0.652795 − 1.13067i 0 −2.77219 + 1.14673i 0
121.17 0 −0.336877 + 1.69897i 0 2.52384 0 −1.81639 + 3.14608i 0 −2.77303 − 1.14469i 0
121.18 0 −0.291910 + 1.70728i 0 0.0833112 0 −1.98056 + 3.43043i 0 −2.82958 − 0.996742i 0
121.19 0 −0.185982 + 1.72204i 0 4.46325 0 1.40757 − 2.43798i 0 −2.93082 − 0.640537i 0
121.20 0 0.130094 + 1.72716i 0 −0.229542 0 1.05145 − 1.82116i 0 −2.96615 + 0.449387i 0
See all 74 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 121.37
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
333.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1332.2.l.b yes 74
3.b odd 2 1 3996.2.l.b 74
9.c even 3 1 1332.2.k.b ✓ 74
9.d odd 6 1 3996.2.k.b 74
37.c even 3 1 1332.2.k.b ✓ 74
111.i odd 6 1 3996.2.k.b 74
333.h even 3 1 inner 1332.2.l.b yes 74
333.l odd 6 1 3996.2.l.b 74
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1332.2.k.b ✓ 74 9.c even 3 1
1332.2.k.b ✓ 74 37.c even 3 1
1332.2.l.b yes 74 1.a even 1 1 trivial
1332.2.l.b yes 74 333.h even 3 1 inner
3996.2.k.b 74 9.d odd 6 1
3996.2.k.b 74 111.i odd 6 1
3996.2.l.b 74 3.b odd 2 1
3996.2.l.b 74 333.l odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{37} - 3 T_{5}^{36} - 112 T_{5}^{35} + 329 T_{5}^{34} + 5664 T_{5}^{33} - 16373 T_{5}^{32} + \cdots + 4304016 \) acting on \(S_{2}^{\mathrm{new}}(1332, [\chi])\). Copy content Toggle raw display